Question

A Carnot engine takes an amount of heat \(Q_{H}=100 .\) J from a high- temperature reservoir at temperature \(T_{H}=1000 .{ }^{\circ} \mathrm{C},\) and exhausts the remaining heat into a low-temperature reservoir at \(T_{\mathrm{L}}=10.0^{\circ} \mathrm{C}\). Find the amount of work that is obtained from this process.

Short answer

Answer: The amount of work obtained from the Carnot engine is 77.76 J.

Step-by-step solution

Convert the given temperatures to Kelvin

In order to work with the temperatures, we first need to convert them from Celsius to Kelvin: \(T_{H} = 1000^{\circ} C + 273.15 = 1273.15 K\) \(T_{L} = 10.0^{\circ}C + 273.15 = 283.15 K\)

Calculate the efficiency of the Carnot engine

Use the formula for the efficiency of a Carnot engine, which is: \(Efficiency = 1 - \frac{T_{L}}{T_{H}}\) Plug in the temperatures calculated in Step 1: \(Efficiency = 1 - \frac{283.15}{1273.15}\) \(Efficiency = 1 - 0.2224\) \(Efficiency = 0.7776\)

Calculate the work obtained from the Carnot engine

Now, use the efficiency and the provided amount of heat, \(Q_H\), to calculate the amount of work, \(W\), obtained from the Carnot engine: \(W = Efficiency \times Q_{H}\) \(W = 0.7776 \times 100\) \(W = 77.76 J\) Therefore, the amount of work obtained from the Carnot engine is \(77.76\) Joules.